Formalized foundations of polynomial real analysisperso.crans.org/~cohen/papers/slides_types2010.pdf · Formalized foundations of polynomial real analysis Cyril Cohen INRIA Saclay

Formalized foundations of polynomial real analysis

Cyril Cohen

INRIA Saclay – Ile-de-FranceLIX Ecole Polytechnique

INRIA Microsoft Research Joint [email protected]

18 octobre 2010

This work has been partially funded by the FORMATH project, nr. 243847, ofthe FET program within the 7th Framework program of the European

Commission.

Cyril Cohen Formalized foundations of polynomial real analysis

Purpose of this formalization

Formalize real polynomial analysis in the SSReflect extension ofthe Coq proof assistant.SSReflect provides a lot of tools and uses a lot of specificprogramming techniques in the domain of finite groups andcombinatorics.Reuse theses techniques to handle more � continuous � theories.


Real Closed Fields

Algebraic structure of reals : Real Closed Fields (RCF)Field + Ordered + Intermediate value theorem for polynomials

x

y

a

f (a)

b

f (b)

c

f (c) = 0


Decidable Equality

In SSReflect, structures have decidable equality.We can define this (implicit) coercion in Coq

Coercion is_true (b : bool) : Prop := (b = true).

SSReflect

uses intensively this coercion

has facilities to go from one point of view to the other(bool-Prop reflection).

We then see boolean equality as propositional equality, for free.


What do we need ?

⇒ Make case analysis on x ≤ y

⇒ Combine statements (using transitivity with both ≤ <,compatibility with operations, etc ..)

⇒ Speak about signs and absolute value

⇒ Use max and min


Taking advantage of the boolean predicate

Making le a boolean predicate.

Like before, consider this boolean predicate as propositionthrough the coercion is_true

⇒ Use equalities to rewrite expressions with order

(x+z <= y+z)= (x <= y)

(sign x == 1)= (0 < x)

. . .

⇒ Use if x <= y then ... else ... in programs


Taking advantage of the boolean predicate

Making le a boolean predicate.

Like before, consider this boolean predicate as propositionthrough the coercion is_true

⇒ Use equalities to rewrite expressions with order

(x+z <= y+z)= (x <= y)

(sign x == 1)= (0 < x)

. . .

⇒ Use if x <= y then ... else ... in programs


Tools

Multiple lemmas about transitivity and compatibility betweenle, lt and field operations

⇒ Need for good naming conventions.


Strict comparison

We’ll define the strict order lt from the large one le by :

Definition lt x y := ~~ (le y x).

and prove its properties.


Ordered ring mixin

Record mixin_of (R : ringType) := Mixin {

le : rel R;

_ : antisymmetric le;

_ : transitive le;

_ : total le;

_ : forall z x y, le x y -> le (x + z) (y + z);

_ : forall x y, le 0 x -> le 0 y -> le 0 (x * y)

}.


Integration in existing SSReflect algebraic hierarchy

Field

IntegralDomain

ComUnitRing

ComRingUnitRing

Ring



Field

IntegralDomain

ComUnitRing

ComRingUnitRing

Ring

OrderedHierarchy



Field

IntegralDomain

ComUnitRing

ComRingUnitRing

Ring

OrderedIntegralDomain

OrderedField


Rolle Theorem for polynomials

A first hint that RCF is a good abstraction of reals :We are able to prove :

Lemma rolle : forall (a b : R) (p : {poly R}),

a < b -> p.[a] = 0 -> p.[b] = 0 ->

exists c, a < c < b /\ p^‘().[c] == 0.

x

y

a

f (a) = 0

b

f (b) = 0

c

f ′(c) = 0


Sketch of the constructive proof

Lemma rolle_weak : forall (a b : R) (p : {poly R}),

a < b -> p.[a] = 0 -> p.[b] = 0 ->

exists c , a < c < b

/\ (p^‘().[c] = 0 \/ p.[c] = 0).

And conclude rolle from it by iterating rolle_weak. Itterminates because P has less than deg(P) roots.


What else can we do with IVT ?

Particularly useful examples

Rolle Theorem

Mean Value Theorem

Write a function that computes the real roots of anypolynomial

Prove that given a polynomial P, and a root x of P, one canfind a neighborhood of x on which P has no root except x .

. . .


Isolation of roots

x

y


Towards quantifier elimination

First step of Quantifier Elimination in RCF.Which entails decidability of the theory of RCF.

Let’s pick one concept from it : Cauchy Index (proof almost done).


Definition of the Cauchy Index

CInd(P

Q, ]a, b[) =

number of positive jumps− number of negative jumps

-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)

−1

0

+1


Useful property of Cauchy Index

PropertyIf P(a),P(b),Q(a),Q(b) 6= 0 then,

CInd

(P

Q, ]a, b[

)+ CInd

(Q

P, ]a, b[

)=

{sign (PQ(b)) if PQ(a)PQ(b) < 0

0 else


Idea of the proof : combinatorics

-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)q(x)/p(x)

−1 0+1



-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)q(x)/p(x)

+1−1 0

+1−1

+1



-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)q(x)/p(x)

+1−1 0

+1−1

+1



-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)q(x)/p(x)

+1−1 0

+1−1

+1

Jumps in the list of signs of PQ. [-1;1;-1;-1;1;-1;1]


Trick of the proof

The sum of jumps of a list l = x0, . . . , xn ∈ {−1, 1}∗ verifies auseful property : it’s the jump between x0 and xn.i.e. {

sign(xn) if x0xn < 0

0 else



-30

-20

-10

0

10

20

30

-4 -2 0 2 4

p(x)/q(x)q(x)/p(x)

+1−1 0

+1−1

+1

Jumps in the list of signs of PQ. [-1;1;-1;-1;1;-1;1]Jump between the first sign -1 and the last one 1, i.e.{

sign (PQ(b)) if PQ(a)PQ(b) < 0

0 else


Conclusion

A library which provides usable tools.It is used in works in progress on

Quantifier elimination in RCF

Formalisation of Bernstein Polynomials


What next ?

Instantiate the Real Closed Fields Structure

Prove some reflexive tactics using it

... to provide a little more automation

Generalize notion of continuity in this context

Extend to further real analysis


The End

Thank you for your attention. Any questions ?


Documents

Formalized foundations of polynomial real analysisperso.crans.org/~cohen/papers/slides_types2010.pdf · Formalized foundations of polynomial real analysis Cyril Cohen INRIA Saclay